Can probability expert solve this question (like at least 1000 answers) Thx!!!!!

Question

Can probability expert solve this question (like at least 1000 answers)

Thx!!!!!!! I just want to learn but I ended up losing my questions from non-experts... Thx again!

Problem A drunk is struggling to walk home. Every time he takes a step he staggers either 1 unit to the right or 1 unit to the left. Every stagger he takes is independent of all his other staggers. He staggers to the right with probability p, and to the left with probability 1 - p. Example: This picture shows the drunk moving along a path that is n 10 steps long. The path shows his position, right or left of his starting point (0,-1,0,-1,-2,-1, 0,-1, 0, 1). After n = 10 steps he is exactly 1 unit to the left of his starting point. a) How far, left or right of his starting position, do you expect him to be after he's taken n steps? Let X be his position after steps

Explanation / Answer

With the ith step of drunkard, define a random variable Xi

Xi= 1 if he takes the step to the right.

   = -1 if he takes the step to the left

Then X =X1+X2+ ………….+ Xn gives the position of the drunkard after n steps.

Define a new variable Yi= ( Xi+1)/2

Then Yi = (1+1/2)= 1 with probability p

              = (-1+1/2)= 0 with probability 1-p

Since the n steps of drunkard are independent.

Yi’s are i.i.d. Bernoulli variates with probability p.

Sum (Yi) ~ B(n,p) then Sum(Xi+1/2) ~ B(n,p) implies X+1/2 ~ B(n,p)

Where

a) To find E(X)

Since E (X+1/2) = np

E(X) +n = 2np

E(X) = n(2p-1)   (Answer)

b) Variance ( X+1)/2) = np(1-p)

      Variance(X) = 4np(1-p)

   S.D.(X) =2 sqrt(np(1-p))

Since the maximum value of E(X) is 0 at p=1/2. Statndard deviation cannot be negative.

Hence at p=1/2 gives maximum value of standard deviation of drunkard after his n steps.

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